A054782 -id:A054782 - cp's OEIS frontend

A277062 Number of primes <= n-th Lucas number.

1, 0, 2, 2, 4, 5, 7, 10, 15, 21, 30, 46, 66, 98, 146, 218, 329, 500, 757, 1158, 1766, 2716, 4164, 6420, 9907, 15320, 23760, 36878, 57356, 89288, 139283, 217506, 340059, 532321, 834147, 1308186, 2053958, 3227229, 5075229, 7987852, 12581575, 19831014

Offset: 0

Vincenzo Librandi, Nov 09 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 0..101 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000032, A000720, A001606, A005479, A052012, A054782.

Magma
```
[#PrimesUpTo(Lucas(n)): n in [0..41]];
```

a:= n-> numtheory[pi]((<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 09 2016

Mathematica
```
Table[PrimePi[LucasL[n]], {n, 0, 50}]
```

a(n) = A000720(A000032(n)). - Michel Marcus, Jun 10 2024

A052011 Number of primes between successive Fibonacci numbers exclusive.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 16, 23, 37, 55, 84, 125, 198, 297, 458, 704, 1087, 1673, 2602, 4029, 6263, 9738, 15186, 23704, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419527, 13298630

Offset: 1

Patrick De Geest, Nov 15 1999

With the given sequence data, we see that neither endpoint is included, so we count primes p in the open interval F(n)Jeppe Stig Nielsen, Jun 06 2015

			Between Fib(9)=34 and Fib(10)=55 we find the following primes: 37, 41, 43, 47 and 53 hence a(9)=5.

Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated from the data at A054782 and A001605)

Cf. A000040, A001605, A005478 (endpoint primes), A010051, A052012, A054782.

a052011 n = a052011_list !! (n-1)
a052011_list = c 0 0 $ drop 2 a000045_list where
  c x y fs'@(f:fs) | x < f     = c (x+1) (y + a010051 x) fs'
                   | otherwise = y : c (x+1) 0 fs
-- Reinhard Zumkeller, Dec 18 2011

for n from 1 to 43 do T[n]:= numtheory:-pi(combinat:-fibonacci(n)) od:
seq(T[n]-T[n-1]-`if`(isprime(combinat:-fibonacci(n)),1,0), n=2..43); # Robert Israel, Jun 08 2015

lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n]+1,Fibonacci[n+1]-1}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
pbf[n_]:=Module[{fib1=If[PrimeQ[Fibonacci[n+1]],PrimePi[Fibonacci[n+1]-1], PrimePi[ Fibonacci[n+1]]], fib0=If[PrimeQ[Fibonacci[n]], PrimePi[ Fibonacci[n]+1],PrimePi[Fibonacci[n]]]},Max[0,fib1-fib0]]; Array[pbf,50] (* Harvey P. Dale, Mar 01 2012 *)

a(n)=my(s); forprime(p=fibonacci(n)+1,fibonacci(n+1)-1,s++); s \\ Charles R Greathouse IV, Jun 08 2015

a(n) = pi(F(n+1)-1) - pi(F(n)) = A000720(A000045(n+1)-1) - A000720(A000045(n)). - Jonathan Vos Post, Mar 08 2010; corrected by Jeppe Stig Nielsen, Jun 06 2015
a(n) ~ phi^(n-1)/(n*sqrt(5)*log(phi)), where phi = (1+sqrt(5))/2 is the golden ratio. - Charles R Greathouse IV, Jun 08 2015
a(n) = A054782(n+1) - A054782(n) - [n+1 in A001605], where [] denotes the Iverson bracket. - Amiram Eldar, Jun 10 2024

A076777 Number of primes between successive Fibonacci numbers inclusive.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 5, 8, 10, 17, 23, 37, 55, 85, 125, 198, 297, 458, 704, 1088, 1673, 2602, 4029, 6263, 9738, 15187, 23704, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419528, 13298630, 21014892, 33227992

Offset: 0

Reinhard Zumkeller, Nov 14 2002

nonn

a(n) = #{p prime | A000045(n)A000045(n+1)}.

			a(10) = 8, as there are 8 primes greater than A000045(10) = 55 and not greater than A000045(10+1) = 89: 59, 61, 67, 71, 73, 79, 83 and 89.

Amiram Eldar, Table of n, a(n) for n = 0..122 (calculated using the b-file at A054782)

Cf. A000045, A000720, A054782, A082602.

Cf. A001605, A005478.

with(combinat): with(numtheory): seq(pi(fibonacci(n+1))-pi(fibonacci(n)),n=0..35); # Emeric Deutsch

Table[PrimePi[Fibonacci[k+1]]-PrimePi[Fibonacci[k]],{k,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 30 2010 *)

A076777(n) = primepi(fibonacci(n+1))-primepi(fibonacci(n))
A076777(n) = sum(i=fibonacci(n)+1,fibonacci(n+1),isprime(i)) \\ Michael B. Porter, Nov 24 2009

a(n) = A000720(A000045(n+1)) - A000720(A000045(n)).

More terms from Emeric Deutsch, Mar 02 2005

More terms from Amiram Eldar, Oct 07 2021

A138184 Largest prime not exceeding Fibonacci(n) = A000045(n).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 53, 89, 139, 233, 373, 607, 983, 1597, 2579, 4177, 6763, 10939, 17707, 28657, 46351, 75017, 121379, 196387, 317797, 514229, 832003, 1346249, 2178283, 3524569, 5702867, 9227443, 14930341, 24157811, 39088157, 63245971

Offset: 3

Colm Mulcahy, Mar 04 2008

			a(8) = 19 because 19 is the largest prime not exceeding 21 = A000045(8).

Harry J. Smith, Table of n, a(n) for n = 3..657

Cf. A138181, A138182, A138183, A138185.

A138184 := proc(n) prevprime(combinat[fibonacci](n)+1) ; end: seq(A138184(n),n=3..45) ; # R. J. Mathar, Apr 22 2008

PrimePrev[n_]:=Module[{k=n},While[ !PrimeQ[k],k-- ];k];f[n_]:=Fibonacci[n];lst={};Do[AppendTo[lst,PrimePrev[f[n]]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 26 2010 *)

a(n) = A000040(A054782(n)). - R. J. Mathar, Apr 22 2008

Edited and extended by R. J. Mathar, Apr 22 2008

Offset changed from 4 to 3 by Harry J. Smith, Jan 02 2009

A082602 Number of primes between successive Fibonacci numbers (including possibly the Fibonacci numbers themselves).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 5, 8, 11, 17, 24, 37, 55, 85, 126, 198, 297, 458, 704, 1088, 1674, 2602, 4029, 6263, 9738, 15187, 23705, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419528, 13298631, 21014892

Offset: 1

Hauke Worpel (hw1(AT)email.com), May 23 2003

nonn

			a(10) = 8 because the 10th Fibonacci number is 55, the 11th is 89 and the eight primes between them are 59, 61, 67, 71, 73, 79, 83 and 89.

Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated using the b-file at A054782)

Cf. A000045, A054782, A076777.

Cf. A001605, A005478.

[#PrimesInInterval(Fibonacci(n-1), Fibonacci(n)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2017

lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n],Fibonacci[n+1]}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)

{ a(n)= c=0; forprime(N=fibonacci(n),fibonacci(n+1),c=c+1); return(c); }

Corrected and extended by Rick L. Shepherd, May 26 2003

a(43)-a(44) from Vincenzo Librandi, Jul 13 2017

A277063 Number of primes <= n-th Bell number.

Original entry on oeis.org

0, 0, 1, 3, 6, 15, 46, 151, 570, 2376, 10961, 54941, 297220, 1720725, 10602541, 69176095, 475881437, 3439093081, 26026621617, 205694058211, 1693554793730, 14494778231067, 128711956613875, 1183763037547762, 11258075170582653, 110558809039675629, 1119649516271861536

Offset: 0

Vincenzo Librandi, Nov 10 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 0..28 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000110, A000720, A054782, A277062.

Magma
```
[#PrimesUpTo(Bell(n)): n in [0..14]];
```
Mathematica
```
Table[PrimePi[BellB[n]], {n, 0, 20}]
```

a(n) = A000720(A000110(n)). - Michel Marcus, Nov 10 2016

a(21)-a(26) from Charles R Greathouse IV, Nov 10 2016

A273974 Number of primes <= n-th Catalan number.

Original entry on oeis.org

0, 0, 1, 3, 6, 13, 32, 82, 226, 651, 1939, 5946, 18637, 59736, 194898, 645742, 2167325, 7359262, 25237989, 87325031, 304549332, 1069685013, 3781189896, 13443608964, 48049822995, 172568075048, 622514907195, 2254799747130, 8197867118026, 29909486953987, 109477635390870

Offset: 0

Vincenzo Librandi, Nov 10 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 0..39 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000108, A000720, A054782, A277062, A277063.

[#PrimesUpTo(Catalan(n)): n in [0..18]];

Table[PrimePi[CatalanNumber[n]], {n, 0, 27}]

a(n) = A000720(A000108(n)). - Michel Marcus, Nov 10 2016

a(28)-a(30) from Amiram Eldar, Sep 03 2024

A352124 Fibonacci numbers k such that pi(k) is also a Fibonacci number.

Original entry on oeis.org

0, 1, 2, 3, 5, 21, 144

Offset: 1

Marc Kouyoumdjian, Mar 05 2022

No examples larger than pi(144) = 34 are known.

Next term is > Fibonacci(123), if it exists (checked using the b-file in A054782). - Amiram Eldar, Mar 05 2022

			21 is a term because 21 = Fibonacci(8) and pi(21) = 8 = Fibonacci(6).

G. L. Honaker, Jr., Prime Curios!.
Ron Knott, The Mathematical Magic of Fibonacci Numbers.

Cf. A000045, A000720, A054782.

Select[(f = Fibonacci[Join[{0}, Range[2, 20]]]), MemberQ[f, PrimePi[#]] &] (* Amiram Eldar, Mar 05 2022 *)

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
lista(nn) = for (n=0, nn, if (n!=1, my(k=fibonacci(n)); if (isfib(primepi(k)), print1(k, ", ")))); \\ Michel Marcus, Mar 07 2022

A153385 Number of primes <= Fibonacci(Fibonacci(n)) = pi(A007570(n)).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 8, 51, 1329, 393790, 5670112879, 43416847208976911

Offset: 0

Harry J. Smith, Dec 25 2008

			a(7) = 51 because Fibonacci(7) = 13, Fibonacci(13) = 233 and there are 51 primes <= 233.

Harry J. Smith, XICalc - Extra Precision Integer Calculator [broken link]
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000045, A000720, A007570, A054782.

[0] cat [#PrimesUpTo(Fibonacci(Fibonacci(n))): n in [1..9]]; // Vincenzo Librandi, Aug 02 2015

PrimePi@# & /@ (Fibonacci@Fibonacci@# & /@ Range@10) (* Robert G. Wilson v, Feb 17 2009 *)

XiCalc
```
Pi(Fib(Fib(n)));
```

a(n) = pi(Fibonacci(Fibonacci(n))) = A000720(A007570(n)).

a(n) = A054782(A000045(n)). - Amiram Eldar, Sep 03 2024

a(11) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 03 2024

A274771 Number of primes <= n-th Carmichael lambda number.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 3, 2, 2, 6, 3, 7, 2, 3, 4, 8, 1, 8, 5, 7, 3, 9, 2, 10, 4, 4, 6, 5, 3, 11, 7, 5, 2, 12, 3, 13, 4, 5, 8, 14, 2, 13, 8, 6, 5, 15, 7, 8, 3, 7, 9, 16, 2, 17, 10, 3, 6, 5, 4, 18, 6, 8, 5, 19, 3, 20, 11, 8, 7, 10, 5, 21, 2

Offset: 1

Vincenzo Librandi, Nov 11 2016

nonn

Cf. A002322, A054782, A070804, A273974, A277062, A277063.

[0] cat [#PrimesUpTo(CarmichaelLambda(n)): n in [2..100]];

with(numtheory):
a:= n-> pi(lambda(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 11 2016

Table[PrimePi[CarmichaelLambda[n]], {n, 100}]

a(n) = A000720(A002322(n)). - Michel Marcus, Nov 11 2016

cp's OEIS Frontend

A277062 Number of primes <= n-th Lucas number.

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A052011 Number of primes between successive Fibonacci numbers exclusive.

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Haskell

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A076777 Number of primes between successive Fibonacci numbers inclusive.

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A138184 Largest prime not exceeding Fibonacci(n) = A000045(n).

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Maple

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A082602 Number of primes between successive Fibonacci numbers (including possibly the Fibonacci numbers themselves).

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Magma

Mathematica

PARI

Extensions

A277063 Number of primes <= n-th Bell number.

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Magma

Mathematica

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A273974 Number of primes <= n-th Catalan number.

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